AI 中文总结
本研究完善了现有张量化形式体系，证明张量化公式具单参数多仿射形式，进而确定了所有可张量化的f-散度，为相关应用提供了完整刻画。
AI 中文摘要
Csiszar提出的f-散度公式引入了一大类用于量化概率分布间差异的泛函。然而，统计学与信息论的诸多应用仅依赖少数f-散度，如Kullback-Leibler散度、χ²-散度及平方Hellinger距离。这些散度尤为实用，因为它们在乘积测度下具有简单的复合公式，该性质有时被称为张量化。本研究对文献中先前提出的张量化形式体系进行了完善，随后证明任何可能的张量化公式都具有由单一参数表征的多仿射形式，并在采用的张量化概念下确定了所有可张量化的f-散度。
英文摘要
Csiszar's formulation of the $f$-divergence introduced a vast family of functionals for quantifying dissimilarity between probability distributions. However, many applications in statistics and information theory rely only on a few $f$-divergences, such as the Kullback-Leibler divergence, the $χ^2$-divergence, and the squared Hellinger distance. These divergences are especially useful because they admit simple compositional formulas under product measures, a property sometimes referred to as tensorization. In this work, we refine a formalism of tensorization previously introduced in the literature. Then, we show that any possible tensorization formula has a multi-affine form characterized by a single parameter, and identify all tensorizable $f$-divergences under our adopted notion of tensorization.
